I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations" (which can be found here: https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160130308).
In the Proof of the second Lemma there is one step in between two lines which I cannot follow (p.466, just before equation (22)), in spite trying to understand it for hours now. I will briefly sum up the problem: Let $|x|< \rho$ and $|\xi|=1$, $w \in L^2(B_{\rho}(0))$. ($d\sigma $ is the surface measure)
Then $$ \int_{0}^{\rho} r^{n-1} dr \int_{|\xi|=1} 1 d\sigma \int^r_0 |w_x (x+ t \xi)| dt$$ $$\leq \int_{r\leq 2 \rho} r^{n-1} dr \int^r_0 \frac{w_x (z)}{|x-z|^{n-1}} dt$$ holds.
So I tried myself and got to this point, using polar coordinates: $$ \int_{|\xi|=1} 1 d\sigma \int^r_0 |w_x (x+ t \xi)| d t = \int_{B_{r}(0)} \frac{|w_x (x+ |z| \xi)|}{|z|^{n-1}} d z$$
I further though if I can find a $\xi$ which minimizes $|w_x (x+ |z| \xi)|$, then that would yield $$|w_x (x+ |z| \xi)| \leq |w_x (x+ |z| \frac{z}{|z|})|=|w_x (x+ z)|$$
From here on though, I don't get any further - I tried substituting in a clever way, but for me it didn't work out. I am in particular confused about the term $\int_{r\leq 2 \rho} r^{n-1} dr$ regarding the larger set over which is integrated. Thank you for any help in advance!
Recall the standard polar coordinate representation on balls, $$ \int_{B_{r}(x)} f(z) \,\mathrm{d}z = \int_0^{r} \int_{S^{n-1}} t^{n-1} f(x + t\xi)\, \mathrm{d}\xi\,\mathrm{d}t.$$ If we apply this with $|f(z)|/|x-z|^{n-1}$ instead of $f,$ noting that $t = |x-z|$ we get, $$ \int_{S^{n-1}} \int_0^r |f(x+t\xi)| \,\mathrm{d}t\,\mathrm{d}\xi = \int_{B_r(x)} \frac{|f(z)|}{|x-z|^{n-1}} \,\mathrm{d}z. $$ Multiplying by $r^{n-1}$ on both sides and integrating on $(0,\rho)$ we get, $$ \int_0^{\rho} r^{n-1} \int_{S^{n-1}} \int_0^r |f(x+t\xi)| \,\mathrm{d}t\,\mathrm{d}\xi\,\mathrm{d}r = \int_0^{\rho} r^{n-1} \int_{B_r(x)} \frac{|f(z)|}{|x-z|^{n-1}} \,\mathrm{d}z\,\mathrm{d}r. $$
This should be sufficient for your purposes, assuming there is a typo in the paper and the $w_x$ should be $|w_x|$ instead. I also think the $2\rho$ is also a typo, as in the next line it's no longer there.